At my school external, I had an 8-year-old student do the same thing! It was interesting. It seemed that he would do this especially if there was an /l/ before or after the /e/. I first targeted the sound in isolation but showing him a drawing of how his tongue is supposed to be in his mouth. I also told him to keep the tip of his tongue behind his bottom teeth while making a "bowl" with the rest of his tongue. Once he got it in isolation, I'd move to the words. He became increasingly successful with the words but unfortunately I wasn't there long enough to move forward.
I'd be interested to know if your student also does this with /l/...it's almost as if he was doing it as a coarticulation (which is why I told him to keep that tongue tip down!). It made him sound like he had an accent but both he and his parents were native English speakers.